I would first like to address a couple of your underlying assumptions:
1) Most "copper" looking coins (in North America anyway) are actually a bronze alloy, reflecting a change from ~95% Cu to ~95% steel. As a result, I don't think that there would be much antimicrobial effect from the coins. Now, if you did really have copper in the water (like copper piping), then the best antimicrobial effects are likely to come from regular stirring/agitation to make sure the bacteria contact the copper.
2) Having some tap water sitting in a warm area is not really a "perfect" breeding ground for bacteria. A perfect breeding ground would contain nutrients (at the very least, sugar). Whatever bacteria you would find is likely the bacteria that circulates in your home air, or other contaminants (like sneezing).
As for your question, how to measure the bacterial load in each bowl, I think the simplest method to do at home would be as @Alan Boyd suggests. Take a small sample of water from each dish, and plate them out on some kind of growth plate (such as a LB agar). After 16-18 hours of growth on the plate, you would just count the number of colonies on each plate, and get a simple measurement of bacterial load from each bowl.
The drawback to this method is that it does not tell you what kind of bacteria are growing, or whether they are likely to cause illness (pathogenic).
For reference, here is a detailed overview of the most common types of agar plates used for microbiology purposes. This is also a protocol for how to make your own plates at home, using readily available gelatin as a gelling agent, rather than using agar (which can be found in some ethnic supermarkets, but is less available).